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\title{Distinguishing disorder from order in irreversible decay processes}
\author{Jonathan W. Nichols, Shane W. Flynn, William E. Fatherley, Jason R. Green\\
Department of Chemistry, University of Massachusetts Boston}
\date{\today}
\maketitle
\section{Introduction}
Rates are a way to infer the mechanism of kinetic processes, such as chemical reactions. They typically obey the empirical mass-action rate laws when the reaction system is homogeneous, with uniform concentration
(s) throughout. Deviations from traditional rate laws are possible when the system is heterogeneous and there are fluctuations in structure, energetics, or concentrations. When traditional kinetic descriptions break down [insert citation], the process is statically and/or dynamically disordered [insert Zwanzig citation], and it is necessary to replace the rate constant in the rate equation with a time-dependent rate coefficient. Measuring the variation of time-dependent rate coefficients is a means of quantifying the fidelity of a rate coefficient and rate law.
In our previous work a theory was developed for analyzing first-order irreversible decay kinetics through an inequality[insert citation]. The usefulness of this inequality is through its ability to quantify disorder, with the unique property of becoming an equality only when the system is disorder free, and therefore described by chemical kinetics in its classical formulation. The next problem that should be addressed is that of higher order kinetics, what if the physical systems one wishes to understand are more complex kinetic schemes, they would require a modified theoretical framework for analysis, but should and can be addressed. To motivate this type of development systems such as...... are all known to proceed through higher ordered kinetics, and all of these systems possess unique and interesting applications, therefore a more complete kinetics description of them should be pursued[insert citations].
Static and dynamic disorder lead to an observed rate coefficient that depends on time $k(t)$. The main result here, and in Reference[cite], is an inequality
\begin{equation}
\mathcal{L}(\Delta{t})^2 \leq \mathcal{J}(\Delta{t})
\end{equation}
between the statistical length (squared)
\begin{equation}
\mathcal{L}(\Delta{t})^2 \equiv \left[\int_{t_i}^{t_f}k(t)dt\right]^2
\end{equation}
and the divergence
\begin{equation}
\frac{\mathcal{J}(\Delta{t})}{\Delta{t}} \equiv \int_{t_i}^{t_f}k(t)^{2}dt
\end{equation}
over a time interval $\Delta t = t_f - t_i$. Both $\mathcal{L}$ and $\mathcal{J}$ are functions of a possibly time-dependent rate coefficient, originally motivated by an adapted form of the Fisher information[cite]. Reference~1 showed that the difference $\mathcal{J}(\Delta t)-\mathcal{L}(\Delta t)^2$ is a measure of the variation in the rate coefficient, due to static or dynamic disorder, for decay kinetics with a first-order rate law. The lower bound holds only when the rate coefficient is constant in first-order irreversible decay. Here we extend this result to irreversible decay processes with ``order'' higher than one. We show $\mathcal{J}-\mathcal{L}^2=0$ is a condition for a constant rate coefficient for any $i$. Accomplishing this end requires reformulating the definition of the time-dependent rate coefficient.
%An inequality between two important quantities known as the statistical length and Fisher divergence is able to quantitatively measure the amount of static and dynamic disorder of a rate coefficient over a period of time, and is able to determine when traditional kinetics is truly valid[insert citation]. This inequality measures the temporal and spatial variation in the rate coefficient. When there is no static or dynamic disorder, this inequality is reduced to an equality and it is the only time where traditional kinetics is valid. It captures the fluctuations associated with the rate coefficient for first order irreversible decay processes.
In this work we extend the application of this inequality to measure disorder in irreversible decay kinetics with nonlinear rate laws (i.e., kinetics with total ``order'' greater thane unity). We illustrate this framework with proof-of-principle analyses for second-order kinetics for irreversible decay phenomena. We also connect this theory to previous work on first-order kinetics showing how the model simplifies in a consistent manner when working with first order models.
\section{Disordered and nonlinear irreversible kinetics}
We consider the irreversible reaction types
\begin{equation}
i\,A \to \mathrm{products}\quad\quad\textrm{for}\quad i=1,2,3,\ldots,n
\end{equation}
with the nonlinear differential rate laws
\begin{equation}
\frac{dC_i(t)}{dt} = k_i(t)\left[C_i(t)\right]^i.
\end{equation}
Experimental data is typically a concentration profile corresponding to the integrated rate law. If the concentration profile is normalized, by dividing the concentration at a time $t$ to the initial concentration, it is called the survival function
\begin{equation}
S_i(t) = \frac{C_i(t)}{C_i(0)},
\end{equation}
the input to our theory. Namely, we define the effective rate coefficient, $k_i(t)$, through an appropriate time derivative of the survival function that depends on the order $i$ of reaction
\begin{equation}
k_i(t) \equiv
\begin{cases}
\displaystyle -\frac{d}{dt}\ln S_1(t) & \text{if } i = 1 \\[10pt]
\displaystyle +\frac{d}{dt}\frac{1}{S_i(t)^{i-1}} & \text{if } i \geq 2.
\end{cases}
\end{equation}
\subsection{Bound for rate constants}
These forms of $k(t)$ satisfy the bound $\mathcal{J}-\mathcal{L}^2 = 0$ in the absence of disorder, when $k_i(t)\to\omega_i$. This is straightforward to show for the case of an $i^{th}$-order reaction ($i\geq 2$), with the traditional integrated rate law
\begin{equation}
\frac{1}{C_i(t)^{i-1}} = \frac{1}{C_i(0)^{i-1}}+(i-1)\omega_i t.
\end{equation}
and associated survival function
\begin{equation}
S_i(t) = \sqrt[i-1]{\frac{1}{1+(i-1)\omega_i tC_i(0)^{i-1}}}.
\end{equation}
In traditional kinetics, the rate coefficient of irreversible decay is assumed to be constant, in which case $k(t)\to\omega_i$, but this will not be the case when the kinetics are statically or dynamically disordered. In these cases, we will use the above definitions of $k(t)$.
The statistical length and divergence can also be derived for these irreversible decay reactions. The time-dependent rate coefficient is
\begin{equation}
k_i(t)
\equiv \frac{d}{dt}\frac{1}{S_i(t)^{i-1}}
= (i-1)\omega_i C_i(0)^{i-1}
\end{equation}
The statistical length $\mathcal{L}_i$ is the integral of the cumulative time-dependent rate coefficient over a period of time $\Delta{t}$, and the divergence is the cumulative square of the rate coefficient, multiplied by the time interval. %The statistical length is
%\begin{equation}
% \mathcal{L}_i(\Delta t)^2 = \left[\int_{t_i}^{t_f}(i-1)\omega_i C_i^{i-1}(0)\,dt\right]^2
%\end{equation}
%and the divergence is
%\begin{equation}
% \frac{\mathcal{J}_i(\Delta t)}{\Delta t} = \int_{t_i}^{t_f}{(i-1)^2\omega_i^2}\left(C_i^{i-1}(0)\right)^{2} dt.
%\end{equation}
For the equations governing traditional kinetics, both the statistical length squared and the divergence are $(i-1)^2\omega_i^2\left(C_i^{i-1}(0)\right)^2\Delta t^2$: the bound holds when there is no static or dynamic disorder, and a single rate coefficient $\omega_i$ is sufficient to characterize irreversible decay.
The nonlinearity of the rate law leads to solutions that depend on concentration. This concentration dependence is also present in both $\mathcal{J}$ and $\mathcal{L}$.
\section{Second-order decay, $A+A\to\textrm{products}$}
Taking our definitions of the integrated rate law, survival function, and time dependent rate coefficient, we are able to apply them to second order irreversible decay. The integrated second-order rate law gives the survival function
%\begin{equation}
% C_A(t) = \frac{C_A(0)}{1+\omega tC_A(0)}
%\end{equation}
\begin{equation}
S_2(t) = \frac{C_2(t)}{C_2(0)} = \frac{1}{1+\omega tC_2(0)}
\end{equation}
and the time-dependent rate coefficient $k_2(t) = \omega_2 C_2(0)$.
$S(t)$ has been changed to fit a second order model of irreversible decay. From this definition of $k(t)$, we define a statistical distance. The statistical distance represents the distance between two different probability distributions, which can be applied to survival functions and rate coefficients.[cite] Integrating the arc length of the survival curve , $\frac{1}{S(t)}$, gives the statistical length
%\begin{equation}
% \mathcal{L}(\Delta{t})^2=\left[\int_{t_i}^{t_f}k(t)dt\right]^2
%\end{equation}
$\mathcal{L}_2(\Delta{t}) = \frac{1}{S(t)}\big|_{S_(t_f)}^{S_(t_i)}$, which measures the cumulative rate coefficient, same as in first order irreversible decay. As seen in first order, the statistical length is also dependent on the time interval, with the statistical length being infinite in an infinite time interval.
In first order irreversible decay, the inequality between the statistical length squared and Fisher divergence determines when a rate coefficient is constant, which is only when the inequality turns into an equality.[cite] A similar inequality is found in second order kinetics.
Putting in the time dependent rate coefficient for a second order irreversible decay, the inequality becomes
\begin{equation}
\omega^2 C_A(0)^2\Delta{t}^2-\left(\omega C_A(0)\Delta{t}\right)^2 \geq 0
\end{equation}
This result is very similar to that of first order irreversible decay($\omega^2\Delta{t}^2)-\left(\omega\Delta{t}\right)^2\geq0$, the only difference being a dependence on the initial concentration of the reactant. This initial concentration dependence serves to cancel the concentration units in the second order rate coefficient, making the statistical length and Fisher divergence dimensionless. When there is a time independent rate coefficient and there is no static disorder, the equality holds $\omega^2 C_A(0)^2\Delta{t}^2 = \left(\omega C_A(0)\Delta{t}\right)^2$.
\section{Second-order kinetic model with dynamical disorder}
To study the effect of dynamical disorder on higher-order kinetics, we adapt the Kohlrausch-Williams-Watts (KWW) model for stretched exponential decay. In first-order kinetics, stretched exponential decay involves a two-parameter survival function $\exp(-\omega t)^\beta$ and associated time-depedent rate coefficient, $k_{1,KWW}(t) = \beta(\omega t)^{\beta}/t$. The parameter $\omega$ is a characteristic rate or inverse time scale and the parameter $\beta$ is a measure of the degree of stretching. We showed in Reference~[citation] that stretching the exponential with $\beta$ increases the degree of dynamic disorder nonlinearly.
In general we can write the $n^{th}$ order rate law for the KWW model using the new formulations for the nonlinear differential KWW and survival function.
\begin{equation}
\frac{d}{dt}[S_n(t)]=\frac{1}{n-1}(\frac{1}{1+z^\beta t^\beta})^{\frac{2-n}{n-1}}(\frac{-z^\beta\beta t^{\beta -1}}{(1+z^\beta t^\beta)^2})
\end{equation}
Such that z contains all the time independent variables $\omega C_a(o)^{n-1}$
Assuming an overall second-order process with a time depedent rate coefficient the survival function is $S_2(t) = 1/\left(1+(\omega tC_2(0))^{\beta}\right)$. We can then simplify the rate law expression.
\begin{equation}
\frac{d}{dt}[S_2(t)]=\frac{-\beta (\omega C_A(0))^\beta t^{\beta -1}}{(1+(\omega C_A(0))^\beta t^\beta)^2}=-k(t)S^2(t)
\end{equation}
The time-depedent rate coefficient characterizing the decay is $k_{2,KWW}(t) = \beta(C_A(0)\omega t)^{\beta}/t$, from the time-derivative of the inverse of the survival function. This definition of the rate coefficient depends on the initial concentration of the reactants which is consistent with units of rate constants in traditional kinetics. Integrating the time-depedent rate coefficient gives the statistical length
\begin{equation}
\mathcal{L}_{KWW}(\Delta{t}) = \left(C_A(0)\omega t\right)^{\beta}\big|_{t_i}^{t_f}
\end{equation}
which also has a concentration dependence. To understand the effect of fluctuations on the rate coefficient we then integrate over the square $k(t)$, determining the divergence.
\begin{equation}
\mathcal{J}_{KWW}(\Delta{t}) =
\frac{\Delta tC_A(0)^{2\beta}\beta^{2}\omega^{2\beta}t^{2\beta-1}}{2\beta-1}\bigg|_{t_i}^{t_f}
\end{equation}
When $\beta=\tfrac{1}{2}$,
\begin{equation}
\mathcal{J}_{KWW}(\Delta{t}) = \Delta t C_A(0)\beta^2\omega^{2\beta} \ln(t)\big|_{t_i}^{t_f}
\end{equation}
We use of this model as a proof-of-principle, however any model a time-depedent rate coefficient can be subject to this analysis.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\columnwidth]{figures/second-order-kww1/second-order-kww1}
\caption{{Plot of $S_{KWW}^{-2}$ for $\beta = 0.25$, $0.5$, $0.75$, and $1.0$.%
}}
\end{center}
\end{figure}
\section{Second-order kinetic model with static disorder}
The difference between the Fisher Divergence and statistical length can also serve as a way of measuring the amount of static disorder in a second order irreversible decay processes. The same model used to show that this is true in first order irreversible decay can be examined again in second order. This model examines two experimentally indistinguishable states A and A', which inter-convert between each other[cite Plonka]. These two states eventually decay irreversibly into B with two time independent rate coefficients $\omega$ and $\omega$' describing the rates of decay. It can be shown that when a distribution of rate coefficients is necessary ($\omega\neq\omega'$), the decay is bi-exponential in second order. When the rate coefficients $\omega$ and $\omega '$ are equal, the decay follows the second order survival function shown in equation 2. In a general case of any order, the survival function is
\begin{equation}
S(t) = \frac{N_A (t)+N_A' (t)}{N_A (0)+N_A' (0)}
\end{equation}
Assuming the states A and A' which decay into B follow second order irreversible decay, the survival function becomes
\begin{equation}
S(t) = \frac{\frac{N_A(0)}{1+\omega tN_A(0)}+\frac{N_A'(0)}{1+\omega'tN_A'(0)}}{N_A (0)+N_A' (0)}
\end{equation}
To simplify and show that the decay is bi-exponential when $\omega\neq\omega'$, the initial concentrations of A and A' are set to be equal. By setting $N_A (0)=N_A' (0)$, the survival function is
\begin{equation}
S(t)=\left(\frac{1}{2+2\omega tN_A(0)}+\frac{1}{2+2\omega'tN_A(0)}\right)
\end{equation}
The survival function in equation 36 shows a bi-exponential decay when $\omega\neq\omega'$, which shows that static disorder is present since a distribution of rate coefficients is necessary to describe the system.
When $\omega=\omega'$, the survival function in equation 37 turns into the standard survival function for a second order reaction shown in equation 12.
\begin{equation}
S(t)=\left(\frac{1}{2+2\omega tN_A(0)}+\frac{1}{2+2\omega tN_A(0)}\right)=\frac{2}{2+2\omega t[A_0]}=\frac{1}{1+\omega t[A_0]}
\end{equation}
\section{Plonka Plots $\frac{1}{S(t)}$ vs time. with changing $\omega/\omega'$}
In the plot of $\frac{1}{S(t)}$ vs. time, the curves intersect each other when $\omega t=1$ at the point $1+[A_0]$\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.17\columnwidth]{figures/scheme/scheme}
\caption{{Kinetic scheme for a model with static disorder. We consider $\lambda= 0$ and treat the decay from each channel as $n^{\textrm{th}}$ order.%
}}
\end{center}
\end{figure}
\section{Conclusions}
The inequality between $\mathcal{L}^2(\Delta{t})$ and $\mathcal{J}(\Delta{t})$ measures not only the amount of static and dynamic order in a first order irreversible decay process, but also in higher order processes. All of these inequalities rely on two functions of the time dependent rate coefficient, the statistical length and divergence. The inequality between the statistical length and Fisher divergence measures the amount of static and dynamic disorder in the rate coefficient. A single rate coefficient is sufficient only when $\mathcal{L}^2(\Delta{t})$=$\mathcal{J}(\Delta{t})$, and is when classical kinetics truly works. This research presents an important discovery and may prove to be very helpful when the order of the reaction is not known. This can be accomplished by fitting data to a survival function and using the proper definition of the time dependent rate coefficient. From that, the difference between the statistical length squared and divergence can be calculated. The smallest difference between the two can determine which order the reaction is. In the future this work may be useful at looking at other kinetic theories such as Michaelis-Menten kinetics.
\section{Acknowledgements}
This material is based upon work supported by the U.S. Army Research Laboratory and the U.S. Army Research Office under grant number W911NF-14-1-0359.
\selectlanguage{english}
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